1. More Object Representations
Parametric instancing
Hierarchical modeling
Constructive Solid Geometry
Sweep Objects
Octrees
Blobs and Metaballs and other such things
Production rules

2. Parametric Instancing
Many things, called primitives, are conveniently described by a label and a few parameters
Cylinder: Radius, length, does it have end-caps, …
Bolts: length, diameter, thread pitch, …
Almost anything you buy from a catalogue
This is a modeling format:
Provide something that knows how to draw the object given the parameters
Depends on the rendering style
Can be an exact representation

3. Rendering Instances
Generally, provide a routine that takes the parameters and produces a polygonal representation
May include texture maps, normal vectors, colors, etc
OpenGL utility library (GLu) defines routines for cubes, cylinders, disks, and other common shapes
Renderman does similar things, so does POVray, …
The procedure may be dynamic
Adjust the polygon resolution according to distance from the viewer

4. OpenGL Support
OpenGL defines display lists for encapsulating commands that are executed frequently
list_id = glGenLists(1);
glNewList(list_id, GL_COMPILE);
glBegin(GL_TRIANGLES);
draw some stuff
glEnd();
glEndList();
And later
glCallList(list_id);

5. Display Lists (2) The list can be:
GL_COMPILE: things don’t get drawn, just stored
GL_COMPILE_AND_EXECUTE: things are drawn, and also stored
The list can contain almost anything:
Useful for encapsulating lighting set-up commands
The list cannot be modified after it is compiled
The whole point is that the list may be internally optimized, which precludes modification
For example, sequences of transformations may be composited into one transformation

6. Display Lists (3) When should you use display lists: Advantages:
When you do the same thing over and over again
Advantages:
Can’t be much slower than the original way
Can be much much faster
Disadvantages:
Doesn’t support real parameterized instancing, because you can’t have any parameters!
Can’t use various commands that would offer other speedups
For example, can’t use glVertexPointer()

7. Hierarchical Modeling
A hierarchical model unites several parametric instances into one object
For example: a desk is made up of many cubes
Generally represented as a tree, with transformations and instances at any node
Rendered by traversing the tree, applying the transformations, and rendering the instances
Particularly useful for rigid-body animation
Human is a hierarchy of body, head, upper arm, lower arm, etc…
Animate by changing the transformations at the nodes

8. Hierarchical Model Example
body
Vitally Important Point:
Every node has its own local coordinate system.
This makes specifying transformations much much easier.
xarm
left arm
rotate about shoulder
upper arm l
translate (l,0,0)
rotate about (0,0,0)
lower arm

9. OpenGL Support OpenGL defines glPushMatrix() and glPopMatrix()
Takes the current matrix and pushes it onto a stack, or pops the matrix off the top of the stack and makes it the current matrix
Note: Pushing does not change the current matrix
Rendering a hierarchy (recursive):
RenderNode(tree)
glPushMatrix()
Apply node transformation
Draw node contents
RenderNode(children)
glPopMatrix()

10. Regularized Set Operations
Hierarchical modeling is not good enough if objects in the hierarchy intersect each other
Transparency will reveal internal surfaces that should not exist
Computing properties like mass counts the same volume twice
Solution is to define regularized set operations:
Just a fancy name
Every object must be a closed volume (mathematically closed)
Define mathematical set operations (union, intersection, difference, complement) to the sets of points within the volume

11. Constructive Solid Geometry (CSG)
Based on a tree structure, like hierarchical modeling, but now:
The internal nodes are set operations: union, intersection or difference (sometimes complement)
The edges of the tree have transformations associated with them
The leaves contain only geometry
Motivated by computer aided design and manufacture
Difference in particular is like drilling or milling
Other operations reduce the number of primitives required
A common format in CAD products

12. CSG Example Fill it in! - - cube   cylinder cylinder scale translate

13. Rendering CSG
Normals and texture coordinates typically come from underlying primitives (not a big deal with CAD)
Some rendering algorithms can render CSG directly
Raytracing (later in the course)
Scan-line with an A-buffer
Can do 2D with tesselators in OpenGL
For OpenGL and other polygon renderers, must convert CSG to polygonal representation
Must remove redundant faces, and chop faces up
Basic algorithm: Split polygons until they are inside, outside, or on boundary. Then choose appropriate set for final answer.
Generally difficult, messy and slow
Numerical imprecision is the major problem

14. CSG Summary Advantages: Disadvantages:
Good for describing many things, particularly machined objects
Better if the primitive set is rich
Early systems used quadratic surfaces
Moderately intuitive and easy to understand
Disadvantages:
Not a good match for polygon renderers
Some objects may be very hard to describe, if at all
Geometric computations are sometimes easy, sometimes hard
A volume representation (hence solid in the name)
Boundary (surface representation) can also work

15. Sweep Objects Define a polygon by its edges Sweep it along a path
The path taken by the edges form a surface - the sweep surface
Special cases
Surface of revolution: Rotate edges about an axis
Extrusion: Sweep along a straight line

16. General Sweeps The path maybe any curve
The polygon that is swept may be transformed as it is moved along the path
Scale, rotate with respect to path orientation, …
One common way to specify is:
Give a poly-line (sequence of line segments) as the path
Give a poly-line as the shape to sweep
Give a transformation to apply at the vertex of each path segment
Difficult to avoid self-intersection

17. Rendering Sweeps Convert to polygons
Break path into short segments
Create a copy of the sweep polygon at each segment
Join the corresponding vertices between the polygons
May need things like end-caps on surfaces of revolution and extrusions
Normals come from sweep polygon and path orientation
Sweep polygon defines one texture parameter, sweep path defines the other

18. Spatial Enumeration
Basic idea: Describe something by the space it occupies
For example, break the volume of interest into lots of tiny cubes, and say which cubes are inside the object
Works well for things like medical data
The process itself, like MRI or CAT scans, enumerates the volume
Data is associated with each voxel (volume element)
Problem to overcome:
For anything other than small volumes or low resolutions, the number of voxels explodes
Note that the number of voxels grows with the cube of linear dimension

19. Octrees (and Quadtrees)
Build a tree where successive levels represent better resolution (smaller voxels)
Large uniform spaces result in shallow trees
Quadtree is for 2D (four children for each node)
Octree is for 3D (eight children for each node)

20. Quadtree Example
top left
top right
bot left
bot right
Octree principle is the same, but there are 8 children

21. Rendering Octrees
Volume rendering renders octrees and associated data directly
A special area of graphics, visualization, not covered in this class
Can convert to polygons by a few methods:
Just take faces of voxels that are on the boundary
Find iso-surfaces within the volume and render those
Typically do some interpolation (smoothing) to get rid of the artifacts from the voxelization
Typically render with colors that indicate something about the data, but other methods exist

22. Spatial Data Structures
Octrees are an example of a spatial data structure
A data structure specifically designed for storing information of a spatial nature
eg Storing the location of fire hydrants in a city
In graphics, octrees are frequently used to store information about where polygons, or other primitives, are located in a scene
Speeds up many computations by making it fast to determine when something is relevant or not
Just like BSP trees speed up visibility
Other spatial data structures include BSP trees, KD-Trees, Interval trees, …